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Asymptotic enumeration and logical limit laws

for expansive multisets and selections

Boris L. Granovsky∗ and Dudley Stark†

May 4, 2005

Abstract

Given a sequence of integers aj, j ≥ 1, a multiset is a combinatorialobject composed of unordered components, such that there are exactlyaj one-component multisets of size j. When aj ≍ jr−1yj for somer > 0, y ≥ 1, then the multiset is called expansive. Let cn be thenumber of multisets of total size n. Using a probabilistic approach, weprove for expansive multisets that cn/cn+1 → 1 and that cn/cn+1 < 1for large enough n. This allows us to prove Monadic Second OrderLimit Laws for expansive multisets. The above results are extendedto a class of expansive multisets with oscillation.

Moreover, under the condition aj = Kjr−1yj+O(yνj), where K >0, r > 0, y > 1, ν ∈ (0, 1), we find an explicit asymptotic formula forcn. In a similar way we study the asymptotic behavior of selectionswhich are defined as combinatorial objects composed of unorderedcomponents of distinct sizes.

∗Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000,Israel, e-mail:mar18aa@techunix.technion.ac.il

†School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS,United Kingdom, e-mail:D.S.Stark@maths.qmul.ac.uk

2000 Mathematics Subject Classification: 6 0C05 (primary), 05A16 (secondary)

1

1 Summary and Historical remarks

Given a sequence of integers aj ≥ 0, j ≥ 1, a multiset is a combinatorialobject of finite total size composed of unordered indecomposable componentssuch that there are exactly aj single component multisets of size j. There is norestriction on the number of times a component may appear in the multiset.Inview of this, let

Ωn =

~η = (η1, η2, . . . , ηn) :

n∑

j=1

jηj = n and ηj ≥ 0 for all j

be the set of unordered integer partitions of an integer n. Any multiset oftotal size n has a component count vector ~η = (η1, η2, . . . , ηn) contained in

Ωn. Here ηj = η(n)j is the number of components of size j in the multiset of

size n considered.(For more details about multisets see e.g.[2, 9]).Let cn be the number of multisets of size n determined by the above

parameters aj , j = 1, . . . , n. We will prove an analytic identity which willbe used to extract information about the growth of cn, as n→ ∞. It followsfrom the definition of a multiset that the number of multisets with a givencomponent count vector ~η = (η1, η2, . . . , ηn) ∈ Ωn is

n∏

j=1

(

aj + ηj − 1

ηj

)

.

Hence, the number of multisets of size n has the Euler type generating func-tion g:

g(x) =

∞∑

n=0

cnxn

= 1 +

∞∑

n=1

∑

~η∈Ωn

n∏

j=1

(

aj + ηj − 1

ηj

)

xjηj

=

∞∏

j=1

(

1− xj)−aj , |x| < 1. (1)

We observe that combinatorial objects that are given by the generating func-tion (1) are also called weighted partitions([8]).

2

The truncated generating function

gn(x) =n∏

j=1

(

1− xj)−aj , |x| < 1 (2)

has Taylor expansion gn(x) =∑∞

k=0 ck,nxk, where ck,n = cn for k ≤ n. For

a fixed n, the series expansion of the function gn(x) converges for all x ∈ C,|x| < 1. We set

x = x(σ, α) = e−σ+2πiα

for some real numbers α and σ. Then we have

∫ 1

0

gn(x)e−2πiαndα =

∫ 1

0

( ∞∑

k=0

ck,ne−kσ+2πiα(k−n)

)

dα = cne−nσ, (3)

where we have used

∫ 1

0

e2πiαmdα =

1, if m = 0,0, if m 6= 0, m ∈ Z.

Substituting (2) into (3) gives the desired identity w.r.t. the free parameterσ ∈ R :

cn = enσ∫ 1

0

n∏

j=1

(

1− e−jσe2πiαj)−aj e−2πiαndα. (4)

A more probabilistic interpretation of (4) can be given. We have

cn = enσn∏

j=1

(

1− e−jσ)−aj

∫ 1

0

n∏

j=1

(

1− e−jσ

1− e−jσe2πiαj

)aj

e−2πiαndα

= enσn∏

j=1

(

1− e−jσ)−aj

∫ 1

0

φ(α) e−2πiαndα, (5)

where φ(α) is given by

φ(α) =

n∏

j=1

φj(α), α ∈ R, (6)

3

for functions φj defined by

φj(α) =

(

1− e−jσ

1− e−jσe2πiαj

)aj

, α ∈ R. (7)

Using the combinatorial identity in Lemma 2.46 in [9] it is easy to see thatfor any σ > 0, φj is the characteristic function of a random variable Xj givenby

P(Xj = jl) =

(

aj + l − 1

l

)

(

1− e−jσ)aj e−ljσ, l = 0, 1, 2, . . . . (8)

Consequently, φ(α) is the characteristic function of Yn :=∑n

j=1Xj, wherethe Xj, j = 1, . . . , n are assumed to be independent. Therefore,

∫ 1

0

φ(α)e−2πiαndα = P (Yn = n) . (9)

Combining (9) with (5) we arrive at the desired representation of cn,which is in the core of the probabilistic method suggested by Khintchine in1950-s ( [20], Chs IV, V) for asymptotic enumeration in the urn models ofstatistical mechanics.The history related to the method is discussed in [15].We note that Khintchine-type representations were subsequently rediscoveredin independent ways by many authors in a variety of seemingly unrelatedcontexts. In particular, observe that (5) can be derived from equation (134) of[3], the latter being based on the conditioning relation (see [2]). In conclusion,the recent paper [22] should be mentioned which treats probabilistic methodsfor enumeration as transforms of generic random variables (in our setting Zj)into specially constructed independent random variables (in our setting Xj).

It follows from (8) that the r.v. j−1Xj is negative binomially distributedwith parameters aj and e−σj , σ > 0. This produces the formula for theexpectation of the sum Yn:

Mn := EYn =n∑

j=1

jaje−jσ

1− e−jσ. (10)

Further on, except for Lemma 1 in Section 2, we will assume that σ = σn > 0is chosen in the unique way so that

Mn = n. (11)

4

The fact that σ can be chosen in such a way follows from observing that Mn

decreases monotonically from ∞ to 0 as σ ranges from 0 to ∞, while n isfixed. The variance of Yn is

B2n := Var(Yn) =

n∑

j=1

j2aje−jσn

(1− e−jσn)2. (12)

We will study the probability in (9) analytically and thereby obtain in-formation about asymptotic behavior of cn, as n→ ∞.

It is natural to suppose that, under some asymptotic conditions on theparameters aj , j ≥ 1, a local limit law should hold for Yn in (9). Asymptoticenumeration of multisets using this approach was apparently initiated in [13],which was preceded by works of Freiman ( see [24]) on the development ofKhintchine’s method. Note that asymptotics of different statistics related tointeger partitions (the case aj = 1, j ≥ 1) was studied by many authors(seee.g.[16, 23]).

In this paper we will initially assume that

aj ≍ jr−1yj, j → ∞, r > 0, y ≥ 1, (13)

where we write aj ≍ bj for sequences aj, bj when there exist constantsD1, D2 > 0 such that D1 ≤ aj/bj ≤ D2 for all j ≥ 1. Although for y > 1,the rate of growth of aj is exponential (but not polynomial) such multisetswill be called, following [5], expansive. This is in view of Bell-Burris Lemma5.2 in [7] which tells us that for y > 1, the asymptotic behavior of the num-ber of multisets with aj ∼ jr−1yj is the same as that of assemblies withaj ∼ jr−1. (Here and in what follows an ∼ bn for sequences an, bn meansthat limn→∞ an/bn = 1).

Provided the parameters aj satisfy (13), we will prove that the normallimiting law for the sum Yn holds, leading to P(Yn = n) ∼ (2πB2

n)−1/2 in (9).

This local limit law, the definition of σn, and (5), will imply our Theorem 1below. A result implying Theorem 1 when in (13) y = 1 and ≍ is replacedby ∼, was obtained by Richmond in [25] and applied for enumeration ofpartitions of n into primes or powers of primes. The first probabilistic proofof the Hardy-Ramanujan formula for partitions ( the case aj = 1, j ≥ 1) wasgiven by Freiman in 1950-s (see Section 2.7 of [24].) Quite recently, a differentprobabilistic proof of this formula was published in [4]. A comprehensivestudy of the asymptotics of integer partitions was made in [?].

5

Theorem 1 appears to be new for y > 1. Note that throughout the paperwe assume, unless it is said otherwise, that all asymptotic expressions arevalid under n→ ∞.

Theorem 1 Assume that (13) holds. Then the number of multisets is asymp-totically

cn ∼ enσn

√

2πB2n

n∏

j=1

(

1− e−jσn)−aj , (14)

where σn is given by (11).

We now formulate an extension of Theorem 1 to a wider than (13) classof parameters aj . Corollary 1 below is an analog of Corollary 1 in [14] forexpansive assemblies.

We write q1(n) •(n) q2(n), if there exist positive constants γ1, γ2,such that γ1q1(n) ≤ •(n) ≤ γ2q2(n), n ≥ 1 . For given 0 < r1 ≤ r2 and y ≥ 1define the set F(r1, r2, y) of parameter functions a = aj , j ≥ 1 satisfying thecondition

jr1−1yj aj jr2−1yj, y ≥ 1, j ≥ 1.

Corollary 1 For an arbitrary r > 0, 0 < ǫ ≤ r/3 and y ≥ 1, the conclusionof Theorem 1 is valid for all parameter functions a ∈ F(2r/3 + ǫ, r, y).

It is interesting to observe that in the case y = 1, our condition a ∈F(2r/3 + ǫ, r, y) implies the condition (i), p. 1084 of Richmond [25]. Thislatter condition is sufficient for the asymptotic formula for partitions of n intoprimes developed in [25]. Generally speaking, multisets with a ∈ F(r1, r2, y)may be called “expansive with oscillation”.

Theorem 1 and Corollary 1 are proved in Section 2. In Section 2 we alsoderive asymptotic estimates for σn and B2

n that are used in (14).A multiset satisfies a monadic second order logical limit law if the proba-

bility that a random representative of the multiset of size n satisfies a monadicsecond order sentence converges, as n→ ∞. Compton [10, 11] showed that toprove that a class of relational structures such as multisets satisfies a monadicsecond order limit law, it suffices to know about the growth of the numberof structures cn of size n. The next corollary from Compton’s theorem wasused in [7] to prove logical limit laws.

6

Theorem 2 [Corollary 8.1 of [7]] Suppose that aj are the parameters ofa multiset such that

cncn+1

∼ y−1 for some y ≥ 1.

If y > 1, then suppose further that there exists N > 0 such that

cncn+1

≤ y−1 whenever n > N.

Then the multiset has a monadic second order logical limit law.

Based on a Tauberian theorem of Schur, Bell and Burris ([7],Theorems 9.1and 9.3) and Bell [6] derived general sufficient conditions on the asymptoticsof aj which imply the hypotheses of Theorem 2. Note that the condition onthe aj obtained in [7] covers the particular case aj ∼ jr−1yj, y ≥ 1 of (13).Combining Theorem 1 with the asymptotic estimates in Section 2, we provein Theorem 3 below the validity of the conditions of Theorem 2, in the case(13) that is not covered by the Bell-Burris sufficient conditions.

Theorem 3 Suppose that aj is a sequence obeying the condition (13). Thenthe corresponding multiset has a monadic second order logical limit law.

A result similar to Theorem 3 is obtained for logarithmic structures in[?].

Moreover, we are able to weaken the condition (13) of Theorem 3:

Corollary 2 For all parameter functions a ∈ F(2r/3 + ǫ, r, y), where r >0, y ≥ 1 and 0 < ǫ ≤ r/3,

cncn+1

= y−1 exp (−δn + o (δn)) ,

where δn = σn − log y → 0.

In particular, Corollary 2 implies that cn/cn+1 → y−1, y ≥ 1, for a ∈F(2r/3+ ǫ, r, 1). A similar result was shown in [14, 15] for certain reversiblecoagulation-fragmentation processes. From an analytical point of view thelatter processes are equivalent to expansive assemblies (see [15]).

7

In view of the above definition, we may consider multisets as unlabelleddecomposable combinatorial structures. We call labelled decomposable com-binatorial objects assemblies, a term used in [3], see also [2]. Sufficient con-ditions to have monadic second order logical limit laws were given for bothmultisets and assemblies in Theorem 6.6 of [10]. Let mj be the number oflabelled components of size j and let aj =

mj

j!. For assemblies the basic

representation (5) becomes for an arbitrary choice of σ

cn = n! enσ exp

(

n∑

j=1

aje−jσ

)

∫ 1

0

ψ(α) e−2πiαndα,

where

ψ(α) = exp

(

n∑

j=1

aje−jσ(e2παj − 1)

)

, r > 0;

see (2.24) of [14] or (125) of [3]. The method of proof of Theorem 3 and thecomment in the last paragraph gives monadic second order logical limit lawsfor assemblies whenever a ∈ F(2r/3 + ǫ, r, 1), r > 0.

Theorem 3 and Corollary 2 are proved in Section 3.The problem of “factorisatio numerorum” can be put in the framework of

enumeration of multisets. The following description of factorisatio numero-rum is taken from [21]. An (additive) arithmetical semigroup is a free commu-tative semigroup G with identity element 1, generated by a countable set Pof “prime” elements, and equipped with an integer-valued “degree”mapping∂ such that

(i) ∂(1) = 0, ∂(p) > 0 for all p ∈ P .

(ii) ∂(ab) = ∂(a) + ∂(b) for all a, b ∈ G.

(iii) The number G#(j) of primes of degree j in G is finite for all integersj.

Multisets can be put into the framework of arithmetical semigroups byletting the operator ∂ stand for the size of the multiset and defining theproduct of two multisets to be their disjoint union. The identity element 1is then just the empty multiset with total size 0.

8

Let f(n) be the total number of unordered factorizations of elementsg ∈ G with ∂(g) = n. Then [21] shows that

∞∑

n=0

f(n)xn =∞∏

j=1

(1− xj)−G#(j), |x| < 1.

This equation is just (1), except that cn has been replaced by f(n) and aj byG#(j). A typical example considered in [21] is polynomials over finite fields,for which G#(j) = qj for some prime q > 1.

We are able to extend the results of [21] and give asymptotic resultsfor “factorisatio numerorum” when aj = G#(j) = Kjr−1yj + O(yνj) forK > 0, ν ∈ (0, 1) and r > 0, y > 1. This involves getting precise enoughestimates of σn in order to derive first order asymptotics of cn. We restrict tothe case y > 1, as then a fairly simple argument using the Poisson summationformula is effective.

Theorem 4 Assume that aj = Kjr−1yj + O (yνj), where K > 0, r > 0,y > 1, and ν ∈ (0, 1). Then cn has asymptotics

cn ∼ κ1ynn−(r+2)/2(r+1) exp

(

κ2nr/(r+1)

)

for positive constants κ1, κ2. Moreover,

κ2 =r + 1

r(KΓ(r + 1))1/(r+1), (15)

where Γ is the gamma function.

For r = 1, (15) recovers the asymptotic formula in [21].Theorem 4 is proved in Section 4.

Remark 1 It is known ([9], p.34) that the generating function g for cn canbe written as

g(x) = expS∗(x), |x| ≤ 1, (16)

where

S∗(x) =

∞∑

j=1

a∗jxj , |x| ≤ 1

9

is the so-called star transformation of the generating function

S(x) =∞∑

j=1

ajxj , |x| ≤ 1

for aj, namely

a∗j =∑

lk=j

alk, j ≥ 1.

The representation (16) says (see e.g. [15]) that g can be viewed also as

a generating function for the parameters a∗j =m∗

j

j!, j ≥ 1 of assemblies. By

Lemma 5.2 in [7], we have that the asymptotic formula in Theorem 4 forenumeration of expansive multisets with y > 1 is also valid for enumerationof assemblies with the same parameters aj. In this connection observe that,under the assumption (13), the orders of the quantities δn, B

2n, ρl(n) found in

Section 2 appear to be the same as the ones in [15] for expansive assemblies.Summing this up, we see that the asymptotic behavior of expansive assembliesand multisets is alike. We will show further on that the same is true also forselections.

We define the selections determined by the parameters aj to be thosemultisets for which no component type appears more than once. For example,if aj = 1 for all j then a selection is an integer partition with distinct parts.Let cn denote the number of selections of size n determined by the aj . Thenthe generating function g for the cn is

g(x) =

∞∏

j=1

(1 + xj)aj , |x| ≤ 1

and analysis similar to the one for multisets gives that in this case j−1Xj is

a binomial r.v. with parameters aj andexp (−jσ)

1+exp (−jσ), where σ > 0 is arbitrary.

Consequently, we have

Theorem 5 Assume that aj satisfy (13). Let σn be chosen in such a waythat

Mn :=

n∑

j=1

jaje−jσn

1 + e−jσn= n.

10

and define Bn by

(Bn)2 =

n∑

j=1

j2aje−jσn

(1 + e−jσn)2. (17)

Then the number of selections is asymptotically

cn ∼ enσn

√

2π(Bn)2

n∏

j=1

(

1 + e−jσn)aj

. (18)

Moreover, if we assume that aj is as in Theorem 4 then cn has the sameasymptotics as cn, with a different constant k1.

We sketch the proof of Theorem 5, which is similar to the proof of The-orems 1 and Theorem 4, in Section 5.

The classic example of an expansive multiset is integer partitions. Forpartitions of an integer, aj = 1 for all j, so that r = y = 1. We can derivethe Hardy-Ramanujan formula giving asymptotics of cn for partitions fromTheorem 1 by using well known properties of the Euler generating functionF (x) :=

∏∞j=1 (1− xj)

−1, |x| < 1. Since

∫∞0

ue−u

1−e−udu = π2

6(see [18], Formula

3.411-7), we apply the Euler-Maclaurin formula (described in detail in [17])to obtain from (11)

n =π2

6σ2n

− 1

2σn+O(1).

Consequently,

σn =π√6n

− 1

4n+O(n−3/2). (19)

The equality (8.6.1) in Section 8.6 of [19] gives

F (e−σn) ∼(σn2π

)1/2

exp

(

π2

6σn

)

. (20)

Finally, Theorem 1 and (20) produce

cn ∼ enσn

√

2πB2n

(σn2π

)1/2

exp

(

π2

6σn

)

. (21)

11

An asymptotic analysis using (19) which we do not present here shows thatthe asymptotic relation (21) still holds when σn is replaced by π/

√6n. Fur-

thermore,

σ3nB

2n ∼

∫ ∞

0

u2e−u

(1− e−u)2du =

π2

3

(see [18], Formula 3.423-3). The above analyis results in the Hardy-Ramanujanformula

cn ∼ eC√n

4n√3

with C = π(2/3)1/2.

Examples of expansive multisets, most of them with r = 1, can be foundin [7],[9]. The simplest example is the class of finite k-colored linear forests,which has aj = kj , so that r = 1, y = k. We give an example with r = 1/2,y = 2.

Example 1 Consider the linear forests in which every tree on j vertices is2-colored with colors red and blue in such a way that it has exactly [j/2] redvertices and j − [j/2] blue vertices. Then aj =

(

j⌊j/2⌋

)

∼√

2/π2jj−1/2.

We may generalize the last example to get multisets with any r ∈ [1/2, 1]as follows.

Example 2 Consider the forests in which every component of size j is com-posed of a linear tree on ⌊jα⌋, α ∈ [0, 1] vertices and a cycle on the j−⌊jα⌋+1vertices, in such a way that one end vertex of the tree is identified with onevertex of the cycle. Call such components(=graphs) lollipops. Suppose weare considering 2-colored lollipops, such that the number of red vertices inthe tree is [jα/2] and the number of blue vertices is jα − [jα/2]. There is norestriction on the number of blue/red vertices in the cycle. The number of2-colored lollipops is

( ⌊jα⌋⌊⌊jα⌋/2⌋

)

2j−⌊jα⌋ ∼√

2/π 2⌊jα⌋/√

⌊jα⌋ × 2j−⌊jα⌋ ∼√

2/π 2jj−α/2.

This example has r = 1− α/2, y = 2.

The next example is a natural case where the multiset satisfies Theorem 3but not the conditions in [6, 7].

12

Example 3 Consider linear forests which are k colored. If a tree is j verticeslong and j is even, then it may be k-colored in all kj possible ways. If j isodd, then the first vertex is always red and the remaining j − 1 vertices maybe colored in all kj−1 possible ways. Then aj = kj if j is even and aj = kj−1

if j is odd. Therefore, (13) holds with r = 1, y = k,D1 = k−1, D2 = 1.

Example 4 Finally, note that r = 2, y = 1 corresponds to plane partitions;see [1].

2 Asymptotics for expansive multisets

In this section we prove Theorem 1 and Corollary 1. Recall that we assumehere, with the exception of Corollary 1, that aj obey the condition (13). Wefirst derive an expansion for the characteristic function φ given by (6) togeneral precision. For any σ > 0, we define the quantities ρl = ρl(n) forl ≥ 3 by

ρl :=n∑

j=1

jlaj

∞∑

k=1

kl−1e−jkσ. (22)

Lemma 1 For a fixed n and any integer s ≥ 3, the function φ can be ex-panded as

φ(α) = exp

(

2πiMnα− 2π2B2nα

2 +

s−1∑

l=3

(2πi)lρll!

αl +O(αsρs)

)

, α → 0,

where Mn and Bn are given by (10) and (12).

Proof The definition (7) implies that for all α ∈ R

φ(α) =n∏

j=1

(

1− e−jσ

1− e−jσne2πiαj

)aj

= exp

(

n∑

j=1

aj

(

log(

1− e−jσ)

− log(

1− e−jσe2πiαj)

)

)

.

13

The logarithms may be expanded in Taylor series as σ > 0 and α ∈ R arefixed, giving

φ(α) = exp

(

n∑

j=1

aj

(

−∞∑

k=1

e−σjk

k+

∞∑

k=1

e−σjke2πijkα

k

))

.

We make use of the Taylor expansion with s ≥ 3

e2πijkα = 1 + 2πijkα− 2π2j2k2α2 +

s−1∑

l=3

(2πijkα)l

l!+O(jsksαs), α→ 0,

which holds uniformly for all j, k ≥ 1, to get

φ(α) = exp

(

n∑

j=1

aj

( ∞∑

k=1

e−jkσ

k

[

2πijkα− 2π2j2k2α2 +s−1∑

l=3

(2πijkα)l

l!+O (αsjsks)

]))

= exp

(

2πi

n∑

j=1

jaje−jσ

1− e−jσnα− 2π2

n∑

j=1

j2aje−jσ

(1− e−jσn)2α2 +

s−1∑

l=3

(2πi)lρll!

αl +O(ρsαs)

)

= exp

(

2πiMnα− 2π2B2nα

2 +s−1∑

l=3

(2πi)lρll!

αl +O(αsρs)

)

, α→ 0. (23)

In what follows we set σ = σn determined by (11) and define δn byδn := σn − log y, y ≥ 1. In proving Theorem 1 we will apply Lemma 1 andso we need estimates of δn, Bn and ρ3. (We use the ability to expand φ(α) tohigher order precision than s = 3 for the proof of Theorem 3 in Section 3.)

Lemma 2 δn ≍ n−1/(r+1), B2n ≍ n(r+2)/(r+1), and ρl(n) ≍ n(r+l)/(r+1), r > 0,

for all l ≥ 3. Moreover, there exists N > 0 such that δn+1 < δn, B2n ≤ B2

n+1,and ρl(n) ≤ ρl(n+ 1) whenever n ≥ N.

Proof We first prove some preliminary facts about δn. Let D1, D2 > 0 beconstants such that D1j

r−1yj ≤ aj ≤ D2jr−1yj, j ≥ 1, y ≥ 1, quadr >

0. Since σn > 0, n ≥ 1, by (11), we deduce from (10),(11) that n ≥∑n

j=1 jaje−jσn ≥ D1

∑nj=1 j

re−jδn, implying that δn > 0 for n large enough.

14

Suppose that there exists a constant ǫ > 0 and a subsequence nk → ∞ suchthat δnk

≥ ǫ. Then, again by (10),(11),

nk ≤ D2

1− y−1e−ǫ

nk∑

j=1

jre−jǫ = O(1), nk → ∞.

We therefore must have δn → 0. Next, we derive from the inequality

n ≥ D1

n∑

j=1

jre−jδn ≥ D1e−nδn

n∑

j=1

jr, r > 0

that nδn → ∞ as n→ ∞.With the help of these facts we further get

n ≤ D2

n∑

j=1

jre−jδn

1− y−je−jδn≤ δ−r−1

n D2

∞∑

j=1

(jδn)re−jδn

1− e−jδnδn ∼ δ−r−1

n D2

∫ ∞

0

xre−x

1− e−xdx.

Since the last integral is bounded, we conclude that δn ≤ D4n−1/(r+1), n ≥

1, where D4 > 0 is a constant. On the other hand,

n ≥ D1

n∑

j=1

jre−jδn ∼ δ−r−1n D1

∫ ∞

0

xre−xdx.

This gives δn ≥ D3n−1/(r+1), n ≥ 1, where D3 > 0 is a constant. We have

shown that δn ≍ n−1/(r+1).For any l ≥ s ≥ 0, arguments similar to those above show that

n∑

j=1

jlaje−jσn

(1− e−jσn)s≍

n∑

j=1

jr+l−1e−jδn

(1− y−je−jδn)s≍ δ−r−l

n ≍ n(r+l)/(r+1). (24)

The last asymptotic applied to (12) results in the stated asymptotics for B2n.

We now show that ρl, l ≥ 3 has the same asymptotics as in (24). Wehave

ρl ≍n∑

j=1

jr+l−1

∞∑

k=1

kl−1e−δnjkyj−jk

∼ δ−r−ln

∞∑

k=1

(

k−r−1

∫ knδn

kδn

zr+l−1yz

kδn(1−k)e−zdz

)

, r ≥ 0.

15

The integral in the last expression is ≤ Γ(r + l) for all k ≥ 1, with equalityfor k = 1, since nδn → ∞, δn → 0, n→ ∞, while y

zkδn

(1−k) ≤ 1, k ≥ 1, y ≥1, z ≥ 0. Thus, the last series in (24) converges which implies

ρl ≍ δ−r−ln ≍ n(r+l)/(r+1). (25)

Lastly we will prove that σn, and so δn, is eventually monotone decreasingin n. Suppose that σn+1 ≥ σn for some n. Then, it would follow that

e−jσn+1

1− e−jσn+1≤ e−jσn

1− e−jσn

for all j ≥ 1, and consequently

n+ 1 =

n+1∑

j=1

jaje−jσn+1

1− e−jσn+1≤

n+1∑

j=1

jaje−jσn

1− e−jσn= n+

(n + 1)an+1y−n−1e−(n+1)δn

1− y−n−1e−(n+1)δn.

In view of the established asymptotics for δn, the last term in the preced-ing inequality tends to 0 for all y ≥ 1. We therefore must have σn+1 < σn forsufficiently large n which implies the same inequality for δn.

The derivative ddx

[e−x/(1− e−x)2] is negative for x ≥ 0, so that if n issufficiently large,

B2n =

n∑

j=1

j2aje−jσn

(1− e−jσn)2<

n+1∑

j=1

j2aje−jσn+1

(1− e−jσn+1)2= B2

n+1

and similarly, ρl(n) ≤ ρl(n+ 1) when n is large.Lemma 3 below proves a local limit theorem for the probability in (9).

Define the sequence α0(n) by

α0(n) := δ(r+2)/2n log n ≍ n−(r+2)/2(r+1) log n. (26)

We will express the integral in (9) as

T = T (n) :=

∫ 1

0

φ(α) e−2πiαndα =

∫ 1/2

−1/2

φ(α) e−2πiαndα = T1(n) + T2(n),

(27)

16

where the middle equality follows from the periodicity of φ(α), as defined by(6), (7), and where

T1 = T1(n) =

∫ α0(n)

−α0(n)

φ(α) e−2πiαndα, (28)

T2 = T2(n) =

∫ −α0(n)

−1/2

φ(α) e−2πiαndα+

∫ 1/2

α0(n)

φ(α) e−2πiαndα. (29)

Lemma 3 and the representation (5) prove Theorem 1.

Lemma 3T1 ∼

(

2πB2n

)−1/2(30)

and for sufficiently large n,

T2 ≤ exp(

−C log2 n)

, (31)

for a constant C > 0, from which it follows that

T =

∫ 1

0

φ(α)e−2πiαn dα ∼(

2πB2n

)−1/2.

Proof The proof of this lemma is similar to the proof of Lemma 6 in [14].Using the expansion of Lemma 1 with s = 3 in the definition (28) of T1 andobserving that, by virtue of Lemma 2, limn→∞ α3ρ3 = 0 and α2B2

n → ∞ forall α ∈ [−α0(n), α0(n)], gives

T1 ∼∫ α0(n)

−α0(n)

exp(

−2π2α2B2n

)

dα ∼(

2πB2n

)−1/2.

The bound for T2 starts with the identity for all α ∈ R,

|φ(α)| =n∏

j=1

∣

∣

∣

∣

1− e−jσn

1− e−jσne2πiαj

∣

∣

∣

∣

aj

= exp

(

−n∑

j=1

aj2log

(

1 +4e−jσn sin2(παj)

(1− e−jσn)2

)

)

.

All of the logarithms are positive, and log(1 + x) ≥ x/(1 + c) whenever0 ≤ x ≤ c for a constant c > 0, so we have

|φ(α)| ≤ exp

−∑

(4σn)−1≤j≤n

aj2log

(

1 +4e−jσn sin2(παj)

(1− e−jσn)2

)

17

≤ exp

−∑

(4σn)−1≤j≤n

C1aje−jσn sin2(παj)

≤ exp

−∑

(4σn)−1≤j≤n

C2jr−1e−jδn sin2(παj)

, α ∈ R,

for some constants C1, C2 > 0. In view of the inequality δn ≤ δn + log y =σn, y ≥ 1, we also have

|φ(α)| ≤ exp

−∑

(4δn)−1≤j≤n

C2jr−1e−jδn sin2(παj)

:= exp(−Vn(α)), α ∈ R.

(32)Since our δn is of the same order as σn in [14], the argument of Lemma 7

in [14], gives the desired estimate of Vn(α):

Vn(α) ≍ δ−rn ≍ nr/(r+1) ≫ log2 n, α ∈ [α0, 1/2] (33)

Proof [of Corollary 1] For a ∈ F(2r/3 + ǫ, r, y), r > 0, 0 < ǫ ≤ r/3, y ≥ 1,the arguments in Lemma 2 show that

n−1/(r1+1) δn n−1/(r2+1), δ−(r1+2)n B2

n δ−(r2+2)n , (34)

δ−(r1+l)n ρl(n) δ−(r2+l)

n for l ≥ 3. (35)

We write α0 = (Bn)−1 logn to obtain, as n→ ∞,

α30ρ3(n) ≤ γ3(log

3 n)δ3(r1+2)/2n δ−(r2+3)

n = γ3(log3 n)δ(3r1−2r2)/2

n → 0,

since 3r1 − 2r2 > 0. We have shown that the asymptotic (30) is still valid.The upper bounds on T2 are like those in the proof of Theorem 1, with thereplacement of (33) by

δ−r1n Vn(α) δ−r2

n for α ∈ [α0, 1/2].

18

3 Logical limit laws for expansive multisets

Lemma 4 below and the asymptotic δn+1 ≍ n−1/(r+1) from Lemma 2 showthat the cn satisfy the hypotheses of Theorem 2 and thereby prove Theorem3.

Lemma 4 If aj ≍ jr−1yj, with r > 0 and y ≥ 1, then

cncn+1

= y−1e−δn+o(δn).

Proof We use (5) to get

cncn+1

= enσn−(n+1)σn+1

n∏

j=1

(

1− e−jσn+1

1− e−jσn

)aj(

1− e−(n+1)σn+1)an+1 T (n)

T (n+ 1),

(36)where T (n) is defined by (27). Since

e−(n+1)σn+1an+1 ≍ (n+ 1)r−1e−(n+1)δn+1 ,

by definition of δn, and

(n+ 1)δn+1 ≍ (n+ 1)r/(r+1),

by Lemma 2, it follows that

(

1− e−(n+1)σn+1)an+1

= eo(δn). (37)

The second factor in the RHS of (36) may be rewritten as

n∏

j=1

(

1− e−jσn+1

1− e−jσn

)aj

= exp

(

n∑

j=1

aj log

(

1− e−jσn+1 − e−jσn

1− e−jσn

)

)

. (38)

We assume that n > N for theN in Lemma 2, so that in particular σn+1 < σn.Since log(1− x) ≤ −x when x ∈ [0, 1], we have

n∏

j=1

(

1− e−jσn+1

1− e−jσn

)aj

≤ exp

(

−n∑

j=1

aje−jσn+1 − e−jσn

1− e−jσn

)

19

≤ exp

(

−n∑

j=1

aje−jσn(jσn − jσn+1)

1− e−jσn

)

= exp

(

−(σn − σn+1)n∑

j=1

jaje−jσn

1− e−jσn

)

= exp (−(σn − σn+1)Mn)

= e−(σn−σn+1)n,

where the second inequality results from the fact that ez − 1 ≥ z, z ≥ 0 .Since log(1− x) ≥ −x/(1 − x) for x ∈ [0, 1], we lower bound (38) by

n∏

j=1

(

1− e−jσn+1

1− e−jσn

)aj

≥ exp

(

−n∑

j=1

aje−jσn+1 − e−jσn

1− e−jσn+1

)

≥ exp

(

−n∑

j=1

aje−jσn+1(jσn − jσn+1)

1− e−jσn+1

)

= exp

(

−(σn − σn+1)n+1∑

j=1

jaje−jσn+1

1− e−jσn+1

)

≥ exp (−(σn − σn+1)Mn+1)

= e−(σn−σn+1)(n+1).

Thus, the product of the first two factors of (36) is bounded above and belowby

e−σn ≤ enσn−(n+1)σn+1

n∏

j=1

(

1− e−jσn+1

1− e−jσn

)aj

≤ e−σn+1 = e−σn+(σn−σn+1).

We bound σn − σn+1 by observing that

1 = (n + 1)− n

=n+1∑

j=1

jaje−jσn+1

1− e−jσn+1−

n∑

j=1

jaje−jσn

1− e−jσn

≥n∑

j=1

jaj(e−jσn+1 − e−jσn)

1− e−jσn

20

≥ (σn − σn+1)n∑

j=1

j2aje−jσn

1− e−jσn,

for n sufficiently large.Thus, recalling that σn − σn+1 > 0, it follows from (24), applied with

l = 2 and s = 1, and Lemma 2 that

σn − σn+1 ≤ O(

δr+2n

)

. (39)

We have shown that

enσn−(n+1)σn+1

n∏

j=1

(

1− e−jσn+1

1− e−jσn

)aj

= exp (−σn + o (δn))

= y−1 exp (−δn + o (δn)) . (40)

Because of (36), (37) and (40), the proof will be completed if we show

that T (n)T (n+1)

= eo(δn). The definitions (27), (28), (29) along with Lemma 2 andLemma 3 imply that

T2(n)

T1(n)= o(δn),

which gives

T (n)

T (n+ 1)= 1 +

T1(n)− T1(n + 1)

T1(n + 1)+ o (δn) . (41)

The definition (28) together with (26) and (39) produce

|T1(n)− T1(n+ 1)| ≤∫ α0(n)

−α0(n)

|φn(α)e−2πiαn − φn+1(α)e

−2πiα(n+1)| dα+ o(δn).

Next, Lemma 1, (11) and the monotonicity of ρl(n) imply that, for asufficiently large fixed n and α→ 0,

φn(α)e−2πiαn − φn+1(α)e

−2πiα(n+1)

= exp(

−2π2α2B2n +Qs(α, n) +O(αsρs(n))

)

(42)

− exp(

−2π2α2B2n+1 +Qs(α, n+ 1) +O(αsρs(n+ 1))

)

= φn(α)e−2πiαn

×

1− exp(

−2π2α2(B2n+1 − B2

n) +Qs(α, n+ 1)−Qs(α, n) +O(αsρs(n + 1)))

,

21

where we denoted

Qs(α, n) =

s−1∑

l=3

(2πi)lρl(n)

l!αl.

We now apply (39), Lemma 2 and (24) with l = 3, s = 2 to get

0 ≤ B2n+1 − B2

n =n+1∑

j=1

j2aje−jσn+1

(1− e−jσn+1)2−

n∑

j=1

j2aje−jσn

(1− e−jσn)2

≤n∑

j=1

j2aje−jσn+1

(1− e−jσn+1)2(jσn − jσn+1) +

(n+ 1)2an+1e−(n+1)σn+1

(1− e−(n+1)σn+1)2

≤ O(

δ−1n

)

. (43)

In a similar way we also have from (22)

0 ≤ ρl(n+ 1)− ρl(n) ≤ (σn − σn+1)

n∑

j=1

jl+1aj

∞∑

k=1

kle−jkσn+1

+ (n+ 1)2l+r−1e−δn(n+1)

≤ O (δn)n∑

j=1

jl+r∞∑

k=1

kle−jkδn+1

= O(

δ−ln

)

n∑

j=1

jr−1

∫ ∞

jδn+1

ule−udu

≤ O(

δ−ln

)

(

∑

j∈D1

jr−1 +∑

j∈D2

jr−1

∫ ∞

nǫ

ule−udu

)

≤ O(

δ−(l+r)n nǫr

)

, l ≥ 3, ǫ > 0. (44)

Here D1 = [1, δ−1n+1n

ǫ] and D2 = [1, n] \ D1. For α ∈ [−α0(n), α0(n)], (26),(43) and Lemma 2 imply

∣

∣(B2n+1 −B2

n)α2∣

∣ ≤ O(δr+1n log2 n) → 0. (45)

Similarly, for α ∈ [−α0(n), α0(n)], l ≥ 3 and all r > 0,

∣

∣(ρl(n + 1)− ρl(n))αl∣

∣ ≤ O(δr( l

2−1)

n nǫr logl n) → 0, (46)

22

for sufficiently small ǫ > 0. Because of (26) and Lemma 2, it follows that forα ∈ [−α0(n), α0(n)],

|αl|ρl(n) = O(

δr(l−2)/2n logl n

)

→ 0, l ≥ 3. (47)

The above discussion reveals the following remarkable feature of the choiceα ∈ [−α0(n), α0(n)] in the expansions (42) and in Lemma 1: under thischoice the terms with s > 3 can be ignored, as n→ ∞. Therefore, based onthe preceding bounds, we get

|φn(α)e−2πiαn − φn+1(α)e

−2πiα(n+1)|≤ O

(

n−1 log2 n)

exp(

−2π2α2B2n +O

(

δr/2n log3 n))

,

uniformly for α ∈ [−α0(n), α0(n)]. Now it follows that

|T1(n)− T1(n+ 1)|T1(n + 1)

≤ O(n−1 log2 n),

and therefore (41) gives T (n)T (n+1)

= 1 + O(n−1 log2 n) + o (δn) = 1 + o (δn) =

exp (o (δn)), proving the lemma.Proof [of Corollary 2] We will make use of (34) and (35). The argumentin the proof of Lemma 2 shows that in the case considered we still have thatδn → 0 as n → ∞ and that δn decreases monotonically for large enough n.Let, as before, r1 = 2r/3 + ǫ, 0 < ǫ ≤ r/3 and r2 = r. Then, observingthat the bound in (39) is valid with r replaced by r1, we get that the boundin (43) becomes O (δr1−r−1

n ). Consequently, setting α0 as before gives (45)with r replaced by r1. The left hand sides of (44) and (47) may be boundedsimilarly.

4 Explicit asymptotic formulae for enumera-

tion of expansive multisets

In this section we will prove Theorem 4, which gives first order asymptoticsfor cn when y > 1, K, r > 0 and

aj = Kjr−1yj +O(yνj), where ν ∈ (0, 1). (48)

23

To approximate σn = log y + δn in the case considered, it is necessary toanalyze the equation

n =

n∑

j=1

jaje−jσn

1− e−jσn

=n∑

j=1

jajy−je−jδn

1− y−je−jδn

=n∑

j=1

jajy−je−jδn +O

(

n∑

j=1

jajy−2je−2jδn

)

=

n∑

j=1

Kjre−jδn +O

(

n∑

j=1

jry−(1−ν)je−jδn

)

+O(1)

=n∑

j=1

Kjre−jδn +O(1).

The Poisson summation formula as used in the proof of Lemma 4 of [12]shows that for l > −1,

n∑

j=1

jle−jδn = Γ(l + 1)δ−l−1n + Cl +O (δn) , (49)

where in the case l > 0 the constant Cl can be found explicitly:

Cl = 2Γ(l + 1)(2π)−l−1ζ(l + 1) cosπ(l + 1)

2

(here ζ(·) is the Riemann zeta function). The preceding estimates imply that

n = Kδ−r−1n Γ(r + 1) +O(1). (50)

from which it follows that

δn =

(

n

KΓ(r + 1)+O(1)

)−1/(r+1)

=

(

n

KΓ(r + 1)

)−1/(r+1)

+ o(n−1)

24

and thatenσn ∼ yn exp

(

nr/(r+1)(KΓ(r + 1))1/(r+1))

.

The asymptotic for B2n follows from

B2n =

n∑

j=1

j2aje−jσn

(1− e−jσn)2

=

n∑

j=1

Kjr+1e−jδn

(1− y−je−jδn)2+

n∑

j=1

j2O(y−(1−ν)j)e−jδn

(1− y−je−jδn)2

=n∑

j=1

Kjr+1e−jδn +O(1)

∼ KΓ(r + 2) δ−r−2n

∼ K−1/(r+1)Γ(r + 1)−(r+2)/(r+1)Γ(r + 2)n(r+2)/(r+1)

The second factor in (14) may be expanded as

n∏

j=1

(

1− e−jσn)−aj = exp

(

n∑

j=1

−aj log(1− y−je−jδn)

)

= exp

(

n∑

j=1

∞∑

k=1

ajy−jke−jkδn

k

)

= exp

(

n∑

j=1

ajy−je−jδn +

n∑

j=1

∞∑

k=2

ajy−jke−jkδn

k

)

.(51)

We use (49) and (50) to show that the first term in the exponential in (51)equals

n∑

j=1

ajy−je−jδn = K

n∑

j=1

jr−1e−jδn +

n∑

j=1

(aj −Kjr−1)y−je−jδn

= KΓ(r)δ−rn +KCr−1 +

∞∑

j=1

y−j(aj −Kjr−1) + o(1)

= Drnr/(r+1) +KCr−1 +

∞∑

j=1

y−j(aj −Kjr−1) + o(1),

25

where

Dr = K1/(r+1)Γ(r)(Γ(r + 1))−r/(r+1) =1

r(KΓ(r + 1)1/(r+1).

The second term in the exponential in (51) equals

n∑

j=1

∞∑

k=2

ajy−jke−jkδn

k=

∞∑

j=1

∞∑

k=2

ajy−jk

k+ o(1),

where the double sum on the right converges absolutely because of (48).

5 Asymptotics for expansive selections

Let c∗n be the number of selections of total size n corresponding to a givensequence aj . The generating function for c∗n is given by

g∗(x) =

∞∑

n=0

c∗nxn

= 1 +

∞∑

n=1

∑

~η∈Ωn

n∏

j=1

(

ajηj

)

xjηj

=

∞∏

j=1

(

1 + xj)aj , |x| < 1.

By adapting the derivation of (4) for multisets to the truncated generatingfunction g∗n(x) =

∏nj=1 (1 + xj)

aj we obtain for all σ ∈ R,

c∗n = enσ∫ 1

0

n∏

j=1

(

1 + e−jσe2πiαj)aj e−2πiαndα. (52)

It follows that

c∗n = enσn∏

j=1

(

1 + e−jσ)aj

∫ 1

0

n∏

j=1

(

1 + e−jσe2πiαj

1 + e−jσ

)aj

e−2πiαndα

= enσn∏

j=1

(

1 + e−jσ)aj

∫ 1

0

φ∗(α) e−2πiαndα, (53)

26

where

φ∗(α) =n∏

j=1

φ∗j(α), α ∈ R,

and

φ∗j (α) =

(

1 + e−jσe2πiαj

1 + e−jσ

)aj

.

If σ > 0, then the φ∗j are characteristic functions of a sequence of independent

binomial random variables j−1Xj :

P(Xk = jl) =

(

akl

)(

e−kσ

1 + e−kσ

)l(1

1 + e−kσ

)ak−l

, l = 0, 1, 2, . . . , ak.

The formula (53) could also be derived from (145) of [3].The number of integer partitions of n with distinct parts all of size at

least s was considered in [13]. This is the selection with

aj =

0 if j < s,1 if j ≥ s.

The identity (53) was derived in [13] for this particular example.Let Y =

∑nj=1 Xj. We have

M∗n := EY =

n∑

j=1

jaje−jσ

1 + e−jσ.

We will assume that σ = σn is chosen in such a way that M∗n = n. The

fact that σn can be chosen in such a way follows from considering that M∗n

decreases from 12

∑nj=1 jaj to 0, as σ changes from 0 to +∞ and noting that

the assumption that the aj satisfy (13) implies that∑n

j=1 jaj > n for n large

enough. Under the above choice of σ, the variance of Y is

(B∗n)

2 := Var(Y ) =

n∑

j=1

j2aje−jσn

(1 + e−jσn)2.

From this starting point the proof of Theorem 5 is similar to the proofsof Theorem 1 and Theorem 4.

27

References

[1] G. Andrews, The Theory of Partitions, Cambridge University Press,1998.

[2] R. A. Arratia, A.Barbour and S. Tavare, Logarithmic combinatorialstructures: A probabilistic approach, EMS monograph in Mathematics,vol.1, European Mathematical Society Publishing house, Zurich, 2003.

[3] R. A. Arratia and S. Tavare, Independent process approximations forrandom combinatorial structures, Adv. Math., 104 (1994), 90 – 154.

[4] L. Baez-Duarte, Hardy-Ramunujan’s asymptotic formula for partitionsand the central limit theorem. Adv. Math., 125, (1997), 114-120.

[5] A. D. Barbour and B. L. Granovsky, Random combinatorial structures:the convergent case,J. Comb. Theory, Ser. A, 109 (2005), 203–220.

[6] J. P. Bell, sufficient conditions for zero-one laws, Trans. Amer. Math.Soc., 354 (2002), 613–630.

[7] J. P. Bell and S. N. Burris, Asymptotics for logical limit laws: when thegrowth of the components is in an RT class, Trans. Amer. Math. Soc.,355 (2003), 3777 – 3794.

[8] N.A. Brigham, On a certain weighted partition function, Proc.Amer.Math. Soc., 1(1950), 192-204.

[9] S. Burris, Number theoretic density and logical limit laws, Mathemati-cal surveys and monographs,86, American Mathematical Society, Prov-idence, RI, 2001.

[10] K. J. Compton, A logical approach to asymptotic combinatorics I. Firstorder properties. Adv. Math. 65 (1987) 65 – 96.

[11] K. J. Compton, A logical approach to asymptotic combinatorics II.Monadic second-order properties. J. Comb. Theory, Ser. A 50 (1989)110 – 131.

28

[12] M. M. Erlihson and B. L. Granovsky, Reversible coagulation-fragmentation processes and random combinatorial structures: asymp-totics for the number of groups,Random sructures and algorithms25(2004), 227-245

[13] G. A. Freiman and J. Pitman, Partitions into distinct large parts, J.Austral. Math. Soc. 57 (1994) 386 – 416.

[14] G. A. Freiman and B. L. Granovsky, Asymptotic formula for a parti-tion function of reversible coagulation-fragmentation processes, Israel J.Math. 130 (2002) 259 – 279.

[15] G. A. Freiman and B. L. Granovsky, Clustering in coagulation-fragmentation processes, random combinatorial structures and additivenumber systems: Asymptotic formulae and limiting laws, Trans. Amer.Math. Soc.357(2005) 2483–2507.

[16] B.Fristedt, The structure of random partitions of large inte-gers,Trans.Amer.Math. Soc.337(1993) 703-735.

[17] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics,Addison-Wesley, 1989.

[18] I. S. Gradshteyn and E. M. Ryzhik, Tables of integrals, series, and prod-ucts, 4th. ed., Academic Press, 1980.

[19] G. H. Hardy. Ramanujan, 3rd ed., Chelsea Publishing Company, 1978.

[20] A. I. Khinchin, Mathematical foundations of quantum statistics, Gray-lock Press, Albany, N.Y., 1960.

[21] A. Knopfmacher, J. Knopfmacher, R. Warlimont, “Factorisatio numero-rum” in arithmetical semigroups, Acta. Arith. 61 (1992) 327 – 336.

[22] O. Milenkovic and K.J. Compton, Probabilistic Transforms for Combi-natorial Urns Models, Comb.,Probab.and Computing13,(2004) 645–675.

[23] L.Mutafchiev, Large distinct part sizes in a random integer partition,Acta Math. Hungar. 87(2000) 47–69.

29

[24] A. G. Postnikov, Introduction to Analytic Number Theory, Translationsof Mathematical Monographs, Vol. 68, American Mathematical Society,Providence, RI, (1987).

[25] L. B. Richmond, A general asymptotic result for partitions, Canad. J.Math. 27 (1975) 1083 – 1091.

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